It appears as the first result but its missing all the key SEO information I've entered using SEO by Yoast (Keyword, Title, Meta Description) and is picking up random info on the page (Twitter feed etc).

Its almost as if Google is not looking at my actual homepage at all.

If I Google the SEO Title of my homepage (Created with SEO by Yoast)
Wedding Band Birmingham West East Midlands Staffordshire Solihull

Google has been known to present different meta information that is presented by the site at times. They do this to best fit the search query. The title and the meta description are your representation to Google and it can choose whether or not to present the data it finds.

There is not a problem with the WordPress SEO by Yoast plugin, your meta information and all details are there, it is a question of whether or not Google decides to use yours or to create one that it feels fits the search query as well as the page that it wants to present. Their goal is to get traffic to your site if it is most relevant and if that means using its own version of title and description then that's what it will do.

There has even been speculation as to some times not even providing a meta description and simply letting Google choose one based on its understanding of the page.

So long answer short, the plugin is functioning 100% correctly and the things you are seeing with Google is what Google is wanting to provide for its results. Your information is there in the code, and the plugin is not responsible for what Google decides to present.

It is a matter of how Google is presenting the information on your homepage. Your homepage is fine, but there is now law or guideline that forces Google to use what you have presented, if they wanted to completely make up a Title and Meta Description for your site and present that, they could.

To my knowledge there is no Re-Examine option with Google, their goal is to make your link in the results page as attractive to click on as possible to make the searcher happy with the results.

What I suggest is not Googling your site url, but rather the name itself Punch The Air, Band or something like that, the site url in a search will give strange results because it is going to have to guess at what you are really looking for.

What is the underlying concern here? People looking for the site are not going to type the url into the search, they will more than likely type the name of the site without the .com .uk .whatever and see what comes up. If they are searching on your url they are doing other research, maybe checking to see if you a legitimate website and at that point the meta and title are a lot less relevant because they are looking for spam or other indicators within.

So what is your main concern? Do you feel that Google is not presenting you highly in searches for certain terms? In the end it is not Yoast or your site configuration, it is how Google is presenting what it finds.

As you can see it brings up no title, and the meta title is just the info on the page itself made up of the first header and part of the text beneath it (And yes I know that my meta description is very similar, but its still not actually showing the meta description).

Unfortunately the randomness is out of any of our control, there is a method to what seems random though. If you look at the first result, yes it only has the Band name, but that is what you are searching for.

On the meta that is shown as the description in place of your actual meta, one thing I see that may be causing that is the similarity between the first paragraph and the meta but also the lack of the band name in the meta description. It appears Google is presenting the "interpreted" meta to highlight the band name and bring attention to that along with the portion of the meta that is matching the first part of the content.
(Might be interesting to add, Punch The Air, in the meta description instead of the "we" and see if that changes the result. There may be some thing to try with how similar the first paragraph and the meta description are currently, I would wonder if having them be less similar would cause the meta description to display more often (just thinking out loud).

Unfortunately there is no way to "force" Google to present one thing or another, but in their defense their goal is never to mislead or misrepresent the page they are displaying. Not that your title and meta do that, but they are and will continue to be in the business of connecting their customers with sites to make them happy and ultimately make them return satisfied Google customers. We encounter pages in our work that have some of the same things happen to them but we understand that what we present is merely a suggestion for Google and that when they change it that this is generally to bring more attention to the listing in some way.

The results for the site look good, they are compelling, and they do match your sites content.